<?xml version="1.0" encoding="UTF-8"?>
<article article-type="Research Article">
  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher">journal-of-information-and-optimization-sciences</journal-id>
      <journal-title-group>
        <journal-title>Journal of Information and Optimization Sciences</journal-title>
      </journal-title-group>
      <issn publication-format="electronic">2169-0103</issn>
      <issn publication-format="print">0252-2667</issn>
      <publisher>
        <publisher-name>Taru Publications</publisher-name>
      </publisher>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.47974/JIOS-1805</article-id>
      <title-group>
        <article-title>Modified Barzilai-Borwein (MBB) gradient method for solving fuzzy nonlinear equations</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes">
          <name>
            <surname>Jardow</surname>
            <given-names>Fanar N.</given-names>
          </name>
          <aff>Department of Environmental Science, College of Environmental Sciences, University of Mosul, Mosul, 41002, Iraq</aff>
        </contrib>
        <contrib contrib-type="author">
          <name>
            <surname>Jameel</surname>
            <given-names>Marwan</given-names>
          </name>
          <aff>Department of Environmental Technologies, College of Environmental Sciences, University of Mosul, Mosul, 41002, Iraq</aff>
        </contrib>
      </contrib-group>
      <volume>46</volume>
      <issue>5</issue>
      <fpage>1559</fpage>
      <lpage>1568</lpage>
      <pub-date date-type="pub">
        <day>15</day>
        <month>07</month>
        <year>2025</year>
      </pub-date>
      <abstract>
        <p>Research has demonstrated that conventional analytical techniques such as the SD, BB, CG, and N approaches are insufficient to solve a collection of fuzzy non-linear equations that involve fuzzy number coefficients[1].  Because of this, the two-step extended gradient method is suggested as an alternative way to resolve these kinds of unclear nonlinear equations. This work uses a six-step technique to solve ambiguous nonlinear equations[7]. This method does not require the Jacobi method and only requires finding a line for t = 0. In order to provide clarification and enhance understanding, two well-known numerical examples are provided to demonstrate the suggested concept, along with their practical results implemented in MATLAB in tables and curves.</p>
      </abstract>
      <kwd-group>
        <kwd>Gradient methods</kwd>
        <kwd>Barzilai-Borwein (BB) method fuzzy numbers</kwd>
        <kwd>Fuzzy non-linear equation</kwd>
      </kwd-group>
      <custom-meta-group>
        <custom-meta>
          <meta-name>access</meta-name>
          <meta-value>open</meta-value>
        </custom-meta>
        <custom-meta>
          <meta-name>retracted</meta-name>
          <meta-value>no</meta-value>
        </custom-meta>
      </custom-meta-group>
    </article-meta>
  </front>
</article>
